Séminaire Lotharingien de Combinatoire, B70j (2014), 50 pp.

Maciej Dołęga, Valentin Féray and Piotr Śniady

Jack Polynomials and Orientability Generating Series of Maps

Abstract.
We study Jack characters,
which are the coefficients of the power-sum expansion of Jack symmetric functions with a suitable normalization.
These quantities have been introduced by Lassalle who formulated some challenging conjectures about them.
We conjecture the existence of a weight on non-oriented
maps (i.e., graphs drawn on non-oriented surfaces)
which allows to express
any given Jack character as a weighted sum of some simple functions indexed by maps.
We provide a candidate for this weight which gives a positive answer to our conjecture in some,
but unfortunately not all, cases.
In particular, it gives a positive answer for Jack characters specialized
to Young diagrams of rectangular shape.
This candidate weight attempts to measure,
in a sense, the non-orientability of a given map.